TITLE: primitive roots and primes
QUESTION [10 upvotes]: Given a positive integer $n > 1$, is it true that there exists infinitely many primes $p$ such that $n$ is a primitive root modulo $p$.

REPLY [11 votes]: It showed up in a recent question so one might wonder. The article Artin's conjecture for primitive roots, Math. Intelligencer, 10 (4) (1988) 59-67 by Ram Murty seems like a good survey. The link is to a dvi copy. It informs one that the result follows from a Generalized Riemann Hypothesis and is unconditionally true for at least one of $2,3,5.$ 
The first $12000$ primes for which $7$ is a primitive root run from $11$ to $378011$. This proportion $\frac{12000}{32141} \approx 0.3734$ agrees well with the theoretical expected proportion of $\prod(1-\frac{1}{p(p-1)}) \approx 0.3739$ where the product is over the primes. The distribution according to congruence class $\mod 7$ is $[1, 1748], [2, 2074], [3, 2032], [4, 2058], [5, 2065], [6, 2023].$ This slight deficit in congruence class 1 seems to hold through this range.